Representations of semigroups by positive linear operators on \(C(X)\) (Q1207703)

Let \(X\) be a compact Hausdorff space, and let \(T\) be a positive linear operator on the Banach lattice \(C(X)\) of all continuous real-valued functions on \(X\). A famous theorem of \textit{G. Choquet} and \textit{C. Foias} [Ann. Inst. Fourier 25, No. 3-4, 109-129 (1976; Zbl 0313.47005)] says: Assume that for each \(x\in X\) there exists \(n\in \mathbb{N}\) such that \((T^ n 1)(x)< 1\). Then \(| T^ n|\to 0\). In the paper under review a theorem of this kind is proved in its most general form: Let \(S\) be a locally compact semitopological semigroup such that for each \(s\in S\) the set \(S\backslash sS\) is relatively compact. Moreover, let \(U: S\to {\mathcal L}(C(X))\) be a bounded representation of \(S\) such that each \(U_ s\) is positive. Assume that for every \(x\in X\) there exists \(s\in S\) satisfying \((U_ s 1)(x)< 1\). Then \(| U_ s|\to 0\) as \(s\to \infty\). Another very interesting result says that weakly almost periodicity of \(\{U_ s: s\in S\}\) implies almost periodicity whenever the action is irreducible and \(S\) and \(U\) satisfy some further mild restrictions. This theorem is a generalization of a result of \textit{H. D. Junghenn} [Semigroup Forum 35, 195-205 (1987; Zbl 0625.43009)].